Incidence (geometry)
Encyclopedia
In geometry
, the relations
of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L1 intersects line L2', in three-dimension
al space). That is, they are the binary relation
s describing how subset
s meet. The propositions of incidence stated in terms of them are statements such as 'any two lines in a plane meet'. This is true in a projective plane
, though not true in Euclidean space
of two dimensions where lines may be parallel
.
Historically, projective geometry
was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of synthetic geometry
it was considered that projective geometry should be developed using such propositions as axiom
s. This turns out to make a major difference only for the projective plane (for reasons to do with Desargues' theorem
).
The modern approach is to define projective space
starting from linear algebra
and homogeneous co-ordinates. Then the propositions of incidence are derived from the following basic result on vector space
s: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P(W) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space (the subspace {0} being common to all subspaces of W), we get the basic proposition of incidence in this form: linear subspace
s L and M of projective space P meet provided dim L + dim M is at least dim P.
where m1 and m2 are slope
s and b1 and b2 are y-intercept
s. Moreover let g be the duality mapping
which maps lines onto their dual points. Then the intersection of lines L1 and L2 is point P3 where
Let g−1 be the inverse duality mapping:
which maps points onto their dual lines. Then the unique line passing through points P1 and P2 is L3 where
the dual of the line is perpendicular
to the point (so that their dot product
is zero); that is, if
where g is the duality mapping.
An equivalent way of checking for this same incidence is to see whether
is true.
If the lines are represented using homogeneous coordinates in the form [m:b:1]L with m being slope and b being the y-intercept, then concurrency can be restated as
Theorem. Three lines L1, L2, and L3 in a projective plane and expressed in homogeneous coordinates are concurrent if and only if their scalar
triple product
is zero, viz. if and only if
Proof. Letting g denote the duality mapping, then
The three lines are concurrent if and only if
According to the previous section, the intersection of the first two lines is a subset of the third line if and only if
Substituting equation (1) into equation (2) yields
but g distributes with respect to the cross product
, so that
and g can be shown to be isomorphic w.r.t. the dot product, like so:
so that equation (3) simplifies to
Q.E.D.
but if the points are expressed in homogeneous coordinates then these three different equations can be collapsed into one equation:
which is more symmetrical and whose computation is straightforward.
If P1 : (x1 : y1 : z1), P2 : (x2 : y2 : z2), and P3 : (x3 : y3 : z3), then P1, P2, and P3 are collinear if and only if
i.e. if and only if the determinant
of the homogeneous coordinates of the points is equal to zero.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the relations
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L1 intersects line L2', in three-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al space). That is, they are the binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
s describing how subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s meet. The propositions of incidence stated in terms of them are statements such as 'any two lines in a plane meet'. This is true in a projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, though not true in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
of two dimensions where lines may be parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
.
Historically, projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
it was considered that projective geometry should be developed using such propositions as axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s. This turns out to make a major difference only for the projective plane (for reasons to do with Desargues' theorem
Desargues' theorem
In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
).
The modern approach is to define projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
starting from linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and homogeneous co-ordinates. Then the propositions of incidence are derived from the following basic result on vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P(W) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space (the subspace {0} being common to all subspaces of W), we get the basic proposition of incidence in this form: linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
s L and M of projective space P meet provided dim L + dim M is at least dim P.
Intersection of a pair of lines
Let L1 and L2 be a pair of lines, both in a projective plane and expressed in homogeneous coordinates:where m1 and m2 are slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
s and b1 and b2 are y-intercept
Y-intercept
In coordinate geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept is a point where the graph of a function or relation intersects with the y-axis of the coordinate system...
s. Moreover let g be the duality mapping
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
which maps lines onto their dual points. Then the intersection of lines L1 and L2 is point P3 where
Determining the line passing through a pair of points
Let P1 and P2 be a pair of points, both in a projective plane and expressed in homogeneous coordinates:Let g−1 be the inverse duality mapping:
which maps points onto their dual lines. Then the unique line passing through points P1 and P2 is L3 where
Checking for incidence of a line on a point
Given line L and point P in a projective plane, and both expressed in homogeneous coordinates, then P⊂L if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the dual of the line is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to the point (so that their dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
is zero); that is, if
where g is the duality mapping.
An equivalent way of checking for this same incidence is to see whether
is true.
Concurrence
Three lines in a projective plane are concurrent if all three of them intersect at one point. That is, given lines L1, L2, and L3; these are concurrent if and only ifIf the lines are represented using homogeneous coordinates in the form [m:b:1]L with m being slope and b being the y-intercept, then concurrency can be restated as
Theorem. Three lines L1, L2, and L3 in a projective plane and expressed in homogeneous coordinates are concurrent if and only if their scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
triple product
Triple product
In mathematics, the triple product is a product of three vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product....
is zero, viz. if and only if
Proof. Letting g denote the duality mapping, then
The three lines are concurrent if and only if
According to the previous section, the intersection of the first two lines is a subset of the third line if and only if
Substituting equation (1) into equation (2) yields
but g distributes with respect to the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
, so that
and g can be shown to be isomorphic w.r.t. the dot product, like so:
so that equation (3) simplifies to
Q.E.D.
Q.E.D.
Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...
Collinearity
The dual of concurrency is collinearity. Three points P1, P2, and P3 in the projective plane are collinear if they all lie on the same line. This is true if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
but if the points are expressed in homogeneous coordinates then these three different equations can be collapsed into one equation:
which is more symmetrical and whose computation is straightforward.
If P1 : (x1 : y1 : z1), P2 : (x2 : y2 : z2), and P3 : (x3 : y3 : z3), then P1, P2, and P3 are collinear if and only if
i.e. if and only if the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the homogeneous coordinates of the points is equal to zero.
See also
- Menelaus theorem
- Ceva's theoremCeva's theoremCeva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively...
- Concyclic
- Incidence matrixIncidence matrixIn mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related ...
- Incidence algebraIncidence algebraIn order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered setand commutative ring with unity.-Definition:...
- Angle of incidenceAngle of incidenceAngle of incidence is a measure of deviation of something from "straight on", for example:* in the approach of a ray to a surface, or* the angle at which the wing or horizontal tail of an airplane is installed on the fuselage, measured relative to the axis of the fuselage.-Optics:In geometric...
- Incidence structureIncidence structureIn mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...
- Incidence geometry
- Levi graphLevi graphIn combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for...
- Hilbert's axiomsHilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
- Incidence (descriptive geometry)